Short-distance digital communication links, between chips on a circuit board, or between different circuit boards for example, have traditionally been built by using electrical interconnects -- metallic tracks and wires. Recent technological advances have resulted in improvements in the speed of information processing, but have left electrical interconnects intact, thus creating a serious communication problem. Free-space optical interconnects, made up of arrays of vertical-cavity surface-emitting lasers, microlenses, and photodetectors, could be used to solve this problem. If free-space optical interconnects are to successfully replace electrical interconnects, they have to be able to support large rates of information transfer with high channel densities. The biggest obstacle in the way of reaching these requirements is laser beam diffraction. There are three approaches commonly used to model the effects of laser beam diffraction in optical interconnects: one could pursue the path of solving the diffraction integral directly, one could apply stronger approximations with some loss of accuracy of the results, or one could cleverly reinterpret the diffraction problem altogether. None of the representatives of the three categories of existing solutions qualified for our purposes. The main contribution of this dissertation consist of, first, formulating the mode expansion method, and, second, showing that it outperforms all other methods previously used for modelling diffraction in optical interconnects. The mode expansion method allows us to obtain the optical field produced by the diffraction of arbitrary laser beams at empty apertures, phase-shifting optical elements, or any combinations thereof, regardless of the size, shape, position, or any other parameters either of the incident optical field or the observation plane. The mode expansion method enables us to perform all this without any reference or use of the traditional Huygens-Kirchhoff-Fresnel diffraction integrals. When using the mode expansion method, one replaces the incident optical field and the diffracting optical element by an effective beam, possibly containing higher-order transverse modes, so that the ultimate effects of diffraction are equivalently expressed through the complex-valued modal weights. By using the mode expansion method, one represents both the incident and the resultant optical fields in terms of an orthogonal set of functions, and finds the unknown parameters from the condition that the two fields have to be matched at each surface on their propagation paths. Even though essentially a numerical process, the mode expansion method can produce very accurate effective representations of the diffraction fields quickly and efficiently, usually by using no more than about a dozen expanding modes. The second tier of contributions contained in this dissertation is on the subject of the analysis and design of microchannel free-space optical interconnects. In addition to the proper characterisation of the design model, we have formulated several optical interconnect performance parameters, most notably the signal-to-noise ratio, optical carrier-to-noise ratio, and the space-bandwidth product, in a thorough and insightful way that has not been published previously. The proper calculation of those performance parameters, made possible by the mode expansion method, was then performed by using experimentally-measured fields of the incident vertical-cavity surface-emitting laser beams. After illustrating the importance of the proper way of modelling diffraction in optical interconnects, we have shown how to improve the optical interconnect performance by changing either the interconnect optical design, or by careful selection of the design parameter values. We have also suggested a change from the usual `square' to a novel `hexagonal' packing of the optical interconnect channels, in order to alleviate the negative diffraction effects. Finally, the optical interconnect tolerance to lateral misalignment, in the presence of multimodal incident laser beams was studied for the first time, and it was shown to be acceptable only as long as most of the incident optical power is emitted in the fundamental Gaussian mode.